# zbMATH — the first resource for mathematics

Analyzing the convergence factor of residual inverse iteration. (English) Zbl 1247.65070
A formula for the convergence factor of the method called residual inverse iteration for nonlinear eigenvalue problems and generalization of the well-known inverse iteration is established. This formula is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows the choice of a vector $$w_k$$ and the formula may be used for the convergence factor so as to analyze the dependence on the choice of $$w_k$$. The formula is also used to illustrate the convergence when the shift is close to the eigenvalue. The slow convergence for double eigenvalues is explained by showing that under generic conditions the convergence factor is one, unless the eigenvalue is semisimple. Convergence similar to the simple case is expected when the eigenvalue is semisimple.

##### MSC:
 65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
NLEVP
Full Text:
##### References:
 [1] Anselone, P., Rall, L.: The solution of characteristic value-vector problems by Newton’s method. Numer. Math. 11, 38–45 (1968) · Zbl 0162.46602 [2] Betcke, M.: Iterative projection methods for symmetric nonlinear eigenvalue problems with applications. Ph.D. thesis, Technical University Hamburg-Harburg (2007) · Zbl 1221.65297 [3] Betcke, M., Voss, H.: Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting. Appl. Math. 52, 267–284 (2007) · Zbl 1164.65412 [4] Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: A collection of nonlinear eigenvalue problems. University of Manchester, MIMS EPrint 2010.98 (2010) · Zbl 1295.65140 [5] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, San Diego (1982) [6] Jarlebring, E., Michiels, W.: Invariance properties in the root sensitivity of time-delay systems with double imaginary roots. Automatica 46, 1112–1115 (2010) · Zbl 1191.93049 [7] Kressner, D.: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114(2), 355–372 (2009) · Zbl 1191.65054 [8] Liao, B.S., Bai, Z., Lee, L.Q., Ko, K.: Nonlinear Rayleigh-Ritz iterative method for solving large scale nonlinear eigenvalue problems. Taiwan. J. Math. 14(3), 869–883 (2010) · Zbl 1198.65072 [9] Meerbergen, K.: The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30(4), 1463–1482 (2008) · Zbl 1176.65041 [10] Mehrmann, V., Voss, H.: Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods. Mitteilungen - Ges. Angew. Math. Mech. 27, 121–152 (2004) · Zbl 1071.65074 [11] Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985) · Zbl 0594.65026 [12] Peters, G., Wilkinson, J.: Inverse iterations, ill-conditioned equations and Newton’s method. SIAM Rev. 21, 339–360 (1979) · Zbl 0424.65021 [13] Rogers, E.: A minimax theory for overdamped systems. Arch. Ration. Mech. Anal. 16, 89–96 (1964) · Zbl 0124.07105 [14] Rott, O., Jarlebring, E.: An iterative method for the multipliers of periodic delay-differential equations and the analysis of a PDE milling model. In: Proceedings of the 9th IFAC Workshop on Time-Delay Systems, Prague, pp. 1–6 (2010) [15] Ruhe, A.: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674–689 (1973) · Zbl 0261.65032 [16] Schreiber, K.: Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. Ph.D. thesis, TU Berlin (2008) · Zbl 1213.65064 [17] Trefethen, L.N., Bau, D.I.: Numerical Linear Algebra. SIAM, Philadelphia (1997) · Zbl 0874.65013 [18] Voss, H.: An Arnoldi method for nonlinear eigenvalue problems. BIT Numer. Math. 44, 387–401 (2004) · Zbl 1066.65059 [19] Voss, H.: Numerical methods for sparse nonlinear eigenvalue problems. In: Proc. XVth Summer School on Software and Algorithms of Numerical Mathematics, Hejnice, Czech Republic (2004). Report 70. Arbeitsbereich Mathematik, TU Hamburg-Harburg · Zbl 1066.65059 [20] Voss, H.: Iterative projection methods for computing relevant energy states of a quantum dot. J. Comput. Phys. 217(2), 824–833 (2006) · Zbl 1102.81040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.